J. Indones. Math. Soc. (MIHMI)
Vol. 14, No. 2 (2008), pp. 95–110.THE EXTREME POSITION AND AMPLITUDE
AMPLIFICATION OF BICHROMATIC WAVES
PROPAGATION BASED ON THIRD ORDER
SOLUTION OF BOUSSINESQ EQUATION
Marwan and Andonowati
Abstract.This paper concerns with the down-stream propagation of waves over initially still water. Such a study is relevant to generate waves of large amplitude in wave tank
of a hydrodynamic laboratory. Input in the form of a time signal is provided at the
wave-maker located at one side of the wave tank; the resulting wave then propagates
over initially still water towards the beach at the other side of the tank. Experiments
show that nonlinear effects will deform the waves and may lead to large wave with waves
height larger than twice the original input; the deformation may show itself as peaking
and splitting. It is direct scientific interest to understand and quantify the nonlinear
distortion; it is also much practical interest to know at which location in the wave tank,
the extreme position, the waves will achieve their maximum amplitude and to know the
amplitude amplification factor. To investigate this, a previously introduced concept called
Maximal Temporal Amplitude (MTA) is used: at each location the maximum over time of
the wave elevation. An explicit expression of the MTA cannot be found in general from the
governing equations and generating signals. In this paper we will use Boussinesq model
and third order approximation theory to calculate the approximate extreme positions
and amplification factor for wave-group that originate from initially bi-chromatic type of
wave. The wave is described by superposition of two monochromatic waves. We show
that, the extreme position depends on the amplitude and the wave length of the group.
The theoretical results are verified with numerical as well as experimental results for
comparison.
Received 11-04-2008, Accepted 27-11-2008.
2000 Mathematics Subject Classification: 35Q35, 35Q53
Key words and Phrases: Boussinesq equation, MTA, wave propagation, asymtotic expansion, AAF, extreme position.
Marwan: Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Syiah Kuala, Jl. Tgk. Daud Bereuh No 4, Darussalam-Banda Aceh 23111, Indonesia.
E-mail: marwan.ramli@math-usk.org
